Enhanced AWG waveform calibration using S-parameters

ABSTRACT

Embodiments of the present invention provide enhanced methods of calibrating arbitrary waveform generators using s-parameters, and arbitrary waveform generators calibrated according to those methods. Methods are provided for calibrating a single, non-interleaved channel of an arbitrary waveform generator, calibrating multiple interleaved channels, and calibrating pairs of channels, both interleaved and non-interleaved, to generate differential signals.

FIELD OF THE INVENTION

The present invention relates to test and measurement instruments, andmore particularly to the calibration of arbitrary waveform generators.

BACKGROUND OF THE INVENTION

Arbitrary Waveform Generators (AWGs) are test and measurementinstruments that are used to generate analog signals having virtuallyany waveshape. In operation, a user defines a desired analog signalpoint-by-point as a series of digital values. An AWG then “plays out”the digital values using a precision digital-to-analog converter toprovide the analog signal. AWGs such as the AWG7000 Arbitrary WaveformGenerator Series available from Tektronix, Inc, of Beaverton, Oreg. areused for wideband signal generation applications, receiver stresstesting of high-speed serial data, and other applications where complexsignal creation is required.

For various reasons, the measured frequency characteristics of signalsproduced by AWGs sometimes differ from the frequency characteristics oftheir input waveform data. Calibration techniques have been proposed tocorrect the output responses of AWGs, however, none of them has provenentirely satisfactory.

Thus, there exists a need for enhanced methods of calibrating AWGs.

SUMMARY OF THE INVENTION

Embodiments of the present invention provide enhanced methods ofcalibrating arbitrary waveform generators using s-parameters, andarbitrary waveform generators calibrated according to those methods.Methods are provided for calibrating a single, non-interleaved channelof an arbitrary waveform generator, calibrating multiple interleavedchannels, and calibrating pairs of channels, both interleaved andnon-interleaved, to generate differential signals.

The objects, advantages, and other novel features of the presentinvention are apparent from the following detailed description when readin conjunction with the appended claims and attached drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 depicts a simplified, high-level block diagram of an arbitrarywaveform generator according to a first embodiment of the presentinvention.

FIG. 2 depicts a first signal flow graph that corresponds to FIG. 1.

FIG. 3 depicts a second signal flow graph that corresponds to FIG. 1.

FIG. 4 depicts a method that corresponds to FIG. 1.

FIG. 5 depicts a simplified, high-level block diagram of an arbitrarywaveform generator according to a second embodiment of the presentinvention.

FIG. 6 depicts a first signal flow graph that corresponds to FIG. 5.

FIG. 7 depicts a second signal flow graph that corresponds to FIG. 5.

FIG. 8 depicts a method that corresponds to FIG. 5.

DETAILED DESCRIPTION OF THE INVENTION

1. Accounting for Reflected Waves

The inventor has recognized that AWGs appear to have imperfect outputresponses because prior AWG calibration techniques have not taken intoaccount the interaction of reflected waves between the AWG and themeasurement instrument during calibration, or between the AWG and thedevice under test (DUT) during use.

Accordingly, embodiments of the present invention provide methods ofcalibrating a channel of an AWG, and arbitrary waveform generatorscalibrated according to those methods, that take into account not onlythe output response of the channel, but also the interaction ofreflected waves between the AWG and a measurement instrument duringcalibration, and between the AWG and the DUT during use.

FIG. 1 depicts an AWG 100 having a single, non-interleaved channelaccording to an embodiment of the present invention. In operation, aprocessor 105 receives waveform data that describes a desired outputanalog signal. The waveform data may be received from a memory, astorage device, or the like. The processor 105 may be implemented assoftware running on a general-purpose microprocessor, a dedicatedapplication specific integrated circuit (ASIC), a field programmablegate array (FPGA), or the like. The processor 105 applies a correctionfilter g to the waveform data in order to correct the output response ofthe channel. The correction filter g can be applied to the waveform databy convolving the correction filter g with the waveform data in the timedomain, or by multiplying them together in the frequency domain. Theprocessed waveform data is converted into an analog signal using adigital-to-analog converter (DAC) 110. The analog signal is filtered byan analog output circuit 115, which may include an amplifier, anattenuator, a switch, a reconstruction filter, and the like. Thefiltered analog signal is then applied to a DUT 120. “The single,non-interleaved channel” refers to the signal path from the DAC 110through the analog output circuit 115. In some embodiments (not shown),the DAC 110 provides a differential output. In that case, the twooutputs may be considered either a pair of channels or a singledifferential channel.

In some embodiments, the correction filter g is calculated as follows:

Referring now to FIG. 2, the output response (amplitude and phase) ofthe channel is measured with a calibrated measurement instrument such asa sampling oscilloscope. The source match, or reflection coefficient, ismeasured with a calibrated measurement instrument such as a time-domainreflectometer (TDR) or a network analyzer. Together, they form thes-parameters of the source S_(21s) and S_(22s). For clarity later in theanalysis, these are written as τ and Γ_(s) respectively,

$\begin{matrix}{\begin{bmatrix}b_{1} \\b_{2}\end{bmatrix} = {\begin{bmatrix}0 & 0 \\\tau & \Gamma_{s}\end{bmatrix}\begin{bmatrix}a_{1} \\a_{2}\end{bmatrix}}} & \left( {{Equation}\mspace{14mu} 1} \right)\end{matrix}$

S_(11s) and S_(12s) equal zero because the input of the DAC is digitalin nature, not analog, and thus, no digital data applied to its inputcan reflect back, and no analog signal applied to its output can passthrough to its input.

To complete the analysis, the DUT input reflection coefficient (Γ_(L))must be known. Then, the response equations can be written:b ₂ =a _(s) gτ+a ₂Γ_(s)  (Equation 2)a ₂ =b ₂Γ_(L)  (Equation 3)

Substituting Equation 3 into Equation 2 yields:b ₂ =a _(s) gτ+b ₂Γ_(L)Γ_(s)  (Equation 4)

Rearranging Equation 4 to solve for b₂ yields:

$\begin{matrix}{b_{2} = \frac{a_{s}g\;\tau}{1 - {\Gamma_{L}\Gamma_{s}}}} & \left( {{Equation}\mspace{14mu} 5} \right)\end{matrix}$

Substituting Equation 5 into Equation 3 yields:

$\begin{matrix}{a_{2} = \frac{a_{s}g\;{\tau\Gamma}_{L}}{1 - {\Gamma_{L}\Gamma_{s}}}} & \left( {{Equation}\mspace{14mu} 6} \right)\end{matrix}$

A. Calibration

A calibrated measurement instrument is defined as an instrument thatcorrectly measures the phase and amplitude of the incoming wave from amatched 50 ohm source. Its input is not necessarily matched, and has aninput reflection coefficient Γ_(L). Similarly, a calibrated AWG isdefined as an AWG that produces an accurate waveform into a matched 50ohm load and has an output reflection coefficient Γ_(s). In that case,Equation 5 reduces to:

$\begin{matrix}{{b_{2\;{match}} = {a_{s}g\;\tau}}{{And}\text{:}}} & \left( {{Equation}\mspace{14mu} 7} \right) \\{{\frac{b_{2\;{match}}}{a_{s}} = 1}{{Therefore}\text{:}}} & \left( {{Equation}\mspace{14mu} 8} \right) \\{g_{match} = \frac{1}{\tau}} & \left( {{Equation}\mspace{14mu} 9} \right)\end{matrix}$

However, when the calibrated instrument and the source are put together(with g=1), the measured result is:

$\begin{matrix}{b_{2\;{meas}} = \frac{a_{s}\tau}{1 - {\Gamma_{L}\Gamma_{s}}}} & \left( {{Equation}\mspace{14mu} 10} \right)\end{matrix}$

Rearranging Equation 10 yields:

$\begin{matrix}{\tau = {\frac{b_{2\;{meas}}}{a_{s}}\left( {1 - {\Gamma_{L}\Gamma_{s}}} \right)}} & \left( {{Equation}\mspace{14mu} 11} \right)\end{matrix}$

Substituting Equation 11 into Equation 9 yields:

$\begin{matrix}{g_{match} = {\frac{1}{\tau} = \frac{a_{s}}{b_{2\;{meas}}\left( {1 - {\Gamma_{L}\Gamma_{s}}} \right)}}} & \left( {{Equation}\mspace{14mu} 12} \right)\end{matrix}$

B. Driving a DUT

Now, when a new device is driven with the calibrated source, theresulting forward wave is:

$\begin{matrix}{b_{2\;{DUT}} = \frac{a_{s}g_{match}\tau}{1 - {\Gamma_{LDUT}\Gamma_{s}}}} & \left( {{Equation}\mspace{14mu} 13} \right)\end{matrix}$

Lastly, the DUT input reflection coefficient must be taken into account.The desired output is just a_(s)g_(match)τ, so the correction filter gis:g=g _(match)(1−Γ_(LDUT)Γ_(s))  (Equation 14)

Where g_(match) represents the correction filter assuming a matchedload, Γ_(LDUT) represents the input reflection coefficient of the DUT,and Γ_(s) represents the output reflection coefficient of the AWG.

In some embodiments, the DUT input reflection coefficient is an ideal,calculated value, selected so that the correction filter corrects theoutput response so that it is right when working into a matched 50 ohmload, an open circuit, or any other specified impedance. This correctionfilter can be generated during manufacturing, stored in the AWG, andused when the DUT s-parameters are not available. In other embodiments,the DUT input reflection coefficient is a measured value by the user, inwhich case the correction filter corrects the output response so that itis right when working into the DUT.

Although the AWG shown and described above only has a single,non-interleaved channel, it will be appreciated that this samecalibration approach can also be used to improve the output response ofan AWG having multiple interleaved channels. That is, the outputresponse of an interleaved AWG can be improved by taking into accountthe interaction of reflected waves between the AWG and the measurementinstrument during calibration, and between the AWG and the DUT duringuse. In that case, the correction filter g developed above can be usedas-is, provided that the multiple interleaved channels are treated as asingle higher rate non-interleaved channel, and the source match of thechannel (Γ_(s)) equals the net source match of the arbitrary waveformgenerator (S_(net)), described in detail below.

C. Adding an External Device

Referring now to FIG. 3, when an external device 125 such as a cable, anup-converter, or the like is used between the AWG 100 and the DUT 120,it can be more appropriate to calibrate at the output of the externaldevice 125. In that case, the correction filter g is essentially thesame as described above. This is because, when the s-parameters of theexternal device 125 are cascaded with the source parameters, the form ofthe new effective source output remains the same because of the twozeros in the first row of the source matrix. The shifted sourceparameters can be measured directly with the external device 125 inplace or calculated using known front panel referenced AWG 100parameters and external device parameters.

FIG. 4 depicts a method 400 of calibrating a channel of an arbitrarywaveform generator according to an embodiment of the present invention.In step 405, an output response of the channel is measured (τ). In step410, a source match of the channel is measured (τ_(s)). In step 415, aninput reflection coefficient of a DUT is determined (τ_(L)). In step420, a correction filter (g) for the channel is calculated based on τ,Γ_(s), and Γ_(L). Steps 405, 410, and 415 are not required to beperformed in the order shown, but rather can be performed in any order.

2. Correcting Multiple Interleaved Channels

Many AWGs achieve higher samples rate by interleaving multiple channelstogether. However, when doing so, the resulting output response is moredifficult to correct for several reasons. The first reason is that theindividual output responses of the interleaved channels will not match,and thus a single correction filter cannot be completely right. Thesecond reason is that the overall output response will be influenced byreflections between the multiple sources, as well as reflections betweenthe multiple sources and the DUT.

Accordingly, embodiments of the present invention provide methods ofcalibrating multiple interleaved channels of an AWG, and arbitrarywaveform generators calibrated according to those methods, that takeinto account the output response of each interleaved channel, theinteraction of reflected waves between the AWG and a measurementinstrument during calibration and between the AWG and the DUT duringuse, or both simultaneously. For reasons that will be explained below,these methods correct the output response of each channel independently,and apply the correction filter to the lower sample rate waveform inputto each DAC rather than the full sample rate waveform.

FIG. 5 depicts an AWG 500 having two interleaved channels according toan embodiment of the present invention. The AWG 500 is similar to theAWG 100, except that it includes two DACs 510A and 510B instead of asingle DAC 110, and a combiner 530. The two DACs 510A and 510B areclocked by two clock signals (not shown) that are phase shifted relativeto one another by 180 degrees. In operation, the processor 505 separatesthe waveform data into samples for the first channel and samples for thesecond channel, and then applies a first correction filter g₁ to thesamples for the first channel, and applies a second correction filter g₂to the samples for the second channel, g₁ and g₂ correct the outputresponses of the first and second interleaved channels, respectively,and also take into account the interaction of reflected waves betweenthe AWG 500 and a measurement instrument during calibration, and betweenthe AWG 500 and the DUT 120 during use. The DAC 510A converts thesamples for the first channel into a first analog signal, and the DAC510B converts the samples for the second channel into a second analogsignal. The first and second analog signals are then combined into asingle analog signal with the combiner 530, which is any device used tocombine analog signals. The resulting analog signal has double thesample rate of either of the individual DACs 510A and 510B. As in theAWG 100, the combined analog signal is then filtered with an analogoutput circuit 115 and applied to a DUT 120. “The first interleavedchannel” refers to the signal path from the processor 505 through theDAC 510A to the analog output circuit 115, and “the second interleavedchannel” refers to the signal path from the processor 505 through theDAC 510B to the analog output circuit 115.

In some embodiments, the correction filters g₁ and g₂ are developed asfollows:

The combiner 530 can be any device used to combine analog signals.However, in the following discussion, the combiner 530 is considered tobe a symmetric, resistive power combiner. Thus, referring now to FIG. 6,the combiner 530 can be represented by a 3×3 s-parameter matrix:

$\begin{matrix}{S = \begin{bmatrix}s_{11} & s_{12} & s_{13} \\s_{21} & s_{22} & s_{23} \\s_{31} & s_{32} & s_{33}\end{bmatrix}} & \left( {{Equation}\mspace{14mu} 15} \right)\end{matrix}$

In matrix notation, the s-parameter equation is:

$\begin{matrix}{{B = {SA}}{{Where}\text{:}}} & \left( {{Equation}\mspace{14mu} 16} \right) \\{{B = \begin{bmatrix}b_{1} \\b_{2} \\b_{3}\end{bmatrix}},{{{and}\mspace{14mu} A} = \begin{bmatrix}a_{1} \\a_{2} \\a_{3}\end{bmatrix}}} & \left( {{Equation}\mspace{14mu} 17} \right)\end{matrix}$

The solution for the output taking into account the source parametersand the combiner is developed in the Appendix.B=(1−SΓ)⁻¹ STA _(s)  (Equation 18)

The difficulty with this approach, however, is that the solutionrequires knowledge of the details of the reflection and transmissionparameters of each channel, along with the two internal ports of thecombiner. However, it is very difficult to directly measure individualparameters once the instrument is assembled. If they can be determinedat all, it would only be through a complex set of calibrationmeasurements and calculations because the response can only be observedat the output.

On the other hand, if the AWG is viewed from the perspective of thesingle output port, then the details of the internal interactions arenot important. This perspective is depicted in FIG. 7, where the outputwave is the sum of the response from each channel to the output, and theDUT 120 interacts with a net single port reflection coefficient at theoutput port. An overall net three port s-parameter network can beconsidered to include the DAC outputs, the interconnect, and thecombiner. With this simplification, Equations 16 and 17 become:

$\begin{matrix}{B^{\prime} = {S_{net}A^{\prime}}} & \left( {{Equation}\mspace{14mu} 19} \right) \\{{B^{\prime} = \begin{bmatrix}b_{1}^{\prime} \\b_{2}^{\prime} \\b_{3}^{\prime}\end{bmatrix}},{{{and}\mspace{14mu} A^{\prime}} = {\begin{bmatrix}a_{1}^{\prime} \\a_{2}^{\prime} \\a_{3}^{\prime}\end{bmatrix} = \begin{bmatrix}{a_{s\; 1}g_{1}} \\{a_{s\; 2}g_{2}} \\a_{3}\end{bmatrix}}}} & \left( {{Equation}\mspace{14mu} 20} \right)\end{matrix}$

The two source ports are idealized; there are no reflections between thesources and the effective combiner, meaning that s₁₁ and s₂₂ are zero;and, returning waves b′₁ and b′₂ are zero, meaning that s₁₂, s₁₃, s₂₁,and s₂₃ are all zero. Thus:

$\begin{matrix}{S_{net} = {\begin{bmatrix}0 & 0 & 0 \\0 & 0 & 0 \\s_{31}^{net} & s_{32}^{net} & s_{33}^{net}\end{bmatrix} = \begin{bmatrix}0 & 0 & 0 \\0 & 0 & 0 \\\tau_{1}^{net} & \tau_{2}^{net} & \Gamma_{3}^{net}\end{bmatrix}}} & \left( {{Equation}\mspace{14mu} 21} \right)\end{matrix}$

Where τ₁ ^(net) and τ₂ ^(net) are the output responses of the twosources measured through the effective combiner. τ₁ ^(net) and τ₂ ^(net)are measured “independently,” that is, the individual output response ofDAC 510A is measured with DAC 510B set to zero, and the individualoutput response of DAC510B is measured with DAC510A set to zero.

We are left with an equation for b₃ dependent on the two source wavesand the reflection from the load, a₃:b ₃=τ₁ ^(net) a _(s1) g ₁+τ₂ ^(net) a _(s2) g ₂+Γ_(s) ^(net) a₃  (Equation 22)a ₃ =b ₃Γ_(L)  (Equation 23)

Substituting Equation 23 into Equation 22 yields:b ₃=τ₁ ^(net) a _(s1) g ₁+τ₂ ^(net) a _(s2) g ₂+Γ_(s) ^(net) b₃Γ_(L)  (Equation 24)

Rearranging yields:

$\begin{matrix}{b_{3} = \frac{{\tau_{1}^{net}a_{s\; 1}g_{1}} + {\tau_{2}^{net}a_{s\; 2}g_{2}}}{1 - {\Gamma_{s}^{net}\Gamma_{L}}}} & \left( {{Equation}\mspace{14mu} 25} \right)\end{matrix}$

Equation 25 is the sum of the response from each channel modified byreflections between the output port and load. It is identical toEquation 5 for the single channel case, except that the sourcetransmission is the sum of two sources.

A. Calibration

Now, when working into a matched load, the output is:b _(3match)=τ₁ ^(net) a _(s1) g ₁+τ₂ ^(net) a _(s2) g ₂  (Equation 26)

If the output from each DAC is measured independently with the other DACset to zero, then the two correction factors are:

$\begin{matrix}{g_{1\;{match}} = {\frac{1}{\tau_{1}^{net}} = \frac{a_{s\; 1}}{b_{3\;{meas}}^{1}\left( {1 - {\Gamma_{L}\Gamma_{s}}} \right)}}} & \left( {{Equation}\mspace{14mu} 27} \right) \\{g_{2\;{match}} = {\frac{1}{\tau_{2}^{net}} = \frac{a_{s\; 2}}{b_{3\;{meas}}^{2}\left( {1 - {\Gamma_{L}\Gamma_{s}}} \right)}}} & \left( {{Equation}\mspace{14mu} 28} \right)\end{matrix}$

Finally, the output of the calibrated source is:

$\begin{matrix}{b_{3} = {\left( {{\tau_{1}^{net}a_{s\; 1}g_{1\;{match}}} + {\tau_{2}^{net}a_{s\; 2}g_{2\;{match}}}} \right)\frac{1}{1 - {\Gamma_{s}^{net}\Gamma_{LDUT}}}}} & \left( {{Equation}\mspace{14mu} 29} \right)\end{matrix}$

B. Driving a DUT

Like the single channel case, if the DUT reflection coefficient isknown, then the source waveform can be compensated to correct for it.This part of the correction can be included in a total filter for eachDAC or applied to the starting waveform at the full sample rate, sinceit is the same for both DACs.

$\begin{matrix}{{b = {\left( {{\tau_{1}^{net}a_{s\; 1}g_{1\;{match}}} + {\tau_{2}^{net}a_{s\; 2}g_{2\;{match}}}} \right)\frac{1}{g_{refl}}}}{{Where}\text{:}}} & \left( {{Equation}\mspace{14mu} 30} \right) \\{g_{refl} = {1 - {\Gamma_{s}^{net}\Gamma_{LDUT}}}} & \left( {{Equation}\mspace{14mu} 31} \right)\end{matrix}$

Thus, the correction filters g₁ and g₂ are as follows:g ₁ =g _(match1) g _(ref1)  (Equation 32)g ₂ =g _(match2) g _(ref1)  (Equation 33)

Where g_(match1) and g_(match2) represent the first and secondcorrection filters assuming a matched load.

Although the discussion above describes generating two correctionfilters for a system having two interleaved channels, it will beappreciated that, by applying similar reasoning, additional correctionfilters can also be developed for systems using higher degrees ofinterleaving. That is, correction filters can be generated for systemshaving three interleaved channels, four interleaved channels, and so on.In that case, to generalize the notation for an arbitrary number ofinterleaved channels, g₁, g₂, and so on are collectively referred to asg_(n), and τ₁ ^(net), τ₂ ^(net), and so on are collectively referred toas τ_(n) ^(net).

Also, although the correction filters described above simultaneouslytake into account both the individual output responses of theinterleaved channels and the effects of reflections between the DUT andthe multiple sources at the same time, correction filters can also begenerated that only take into account the individual output responses ofthe interleaved channels. That is, in some embodiments, an arbitrarywaveform generator is calibrated by measuring the output response ofeach interleaved channel independently and then generating a pluralityof correction filters, one for each interleaved channel, based solely onits corresponding measured output response. In that case, eachcorrection filter equals the inverse of its associated measured outputresponse. In other embodiments, the DUT input reflection coefficient andthe net source match of the AWG are also measured and used to improvethe accuracy of those correction filters.

FIG. 8 depicts a method 800 of calibrating a plurality of interleavedchannels of an arbitrary waveform generator according to an embodimentof the present invention. In step 805, a plurality of output responses,one for each of the plurality of interleaved channels is measured (τ_(n)^(net)). Optionally, in step 810, a net source match of an output portof the arbitrary waveform generator is measured (S_(net)). Optionally,in step 815, an input reflection coefficient of a device under test isdetermined (Γ_(L)). In step 820, a plurality of correction filters(g_(n)) are calculated, one for each of the plurality of interleavedchannels, based on τ_(n) ^(net), S_(net), and Γ_(L). Steps 805, 810, and815 are not required to be performed in the order shown, but rather canbe performed in any order.

3. Correcting Pairs of Channels used to Generate Differential Signals

In some cases, pairs of channels are used to generate differentialsignals, both pairs of single, non-interleaved channels and pairs ofmultiple interleaved channels. Each of those channels can beindividually calibrated using the techniques described above.Alternatively, the pairs of channels can be calibrated simultaneouslyusing the techniques described above by replacing the single-endedparameters with differential parameters. That is, the single-endedoutput response of a channel (τ) would be replaced with the differentialoutput response of a pair of single-ended, non-interleaved channels, orthe differential output response of a pair of multiple interleavedchannels, and so on.

It will be appreciated from the foregoing discussion that the presentinvention represents a significant advance in the field of test andmeasurement instruments. Although specific embodiments of the inventionhave been illustrated and described for purposes of illustration, itwill be understood that various modifications may be made withoutdeparting from the spirit and scope of the invention. Accordingly, theinvention should not be limited except as by the appended claims.

APPENDIX

As defined above, the combiner is represented with a 3 port s-parametermatrix:

$\begin{matrix}{{B = {SA}}{{Where}\text{:}}} & \left( {{Equation}\mspace{14mu} 34} \right) \\{{B = \begin{bmatrix}b_{1} \\b_{2} \\b_{3}\end{bmatrix}},{{{and}\mspace{14mu} A} = \begin{bmatrix}a_{1} \\a_{2} \\a_{3}\end{bmatrix}}} & \left( {{Equation}\mspace{14mu} 35} \right)\end{matrix}$

Given reflection coefficients for the two channels and the load, theelements of A are:a ₁ =a _(s1) g ₁τ₁+Γ_(s1) b ₁  (Equation 36)a ₂ =a _(s2) g ₂τ₂+Γ_(s2) b ₂  (Equation 37)a ₃=Γ_(L) b ₃  (Equation 38)

Then, Equation 34 can be written:

$\begin{matrix}{{B = {{STA}_{s} + {S\;\Gamma\; B}}}{{Where}\text{:}}} & \left( {{Equation}\mspace{14mu} 39} \right) \\{{{\Gamma = \begin{bmatrix}\Gamma_{s\; 1} & 0 & 0 \\0 & \Gamma_{s\; 2} & 0 \\0 & 0 & \Gamma_{L}\end{bmatrix}},{T = \begin{bmatrix}\tau_{1} & 0 & 0 \\0 & \tau_{2} & 0 \\0 & 0 & \tau_{L}\end{bmatrix}},{and}}{A_{s} = \begin{bmatrix}{a_{s\; 1}g_{1}} \\{a_{s\; 2}g_{2}} \\a_{3}\end{bmatrix}}} & \left( {{Equation}\mspace{14mu} 40} \right)\end{matrix}$

Next, rearrange to solve for B:(1−SΓ)B=STA,  (Equation 41)B=(1−SΓ)⁻¹ STA _(s)  (Equation 42)

Equation 42 can be written as a simple matrix equation with (1−SΓ)⁻¹S=KB=KA _(s)  (Equation 43)And then:A=S ⁻¹ KA _(s)  (Equation 44)

Equation 43 solves for all three terms in B, but b₃ is the one we areinterested in. From Equations 40 and 43 the solution for b₃ is:b ₃ =k ₃₁ a _(s1) g ₁ +k ₃₂ a _(s2) g ₂ +k ₃₃ a ₃  (Equation 45)

But a₃ is just:a ₃ =b ₃Γ_(L)  (Equation 46)

Which can be substituted into and Equation 43 and solved for b₃.

Equation 43 is not quite an s-parameter equation since b₁ and b₂ areinternal and not at the ports where a_(s1) and a_(s2) are defined.However, it suggests that a net effective s-parameter equation can bewritten.

What is claimed is:
 1. A method of calibrating a plurality of interleaved channels of an arbitrary waveform generator comprising the steps of: measuring a plurality of output responses, one for each of the plurality of interleaved channels (τ_(n) ^(net)); measuring a net source match of an output port of the arbitrary waveform generator (S_(net)); determining an input reflection coefficient of a device under test (Γ_(L)); calculating a plurality of correction filters (g_(n)), one for each of the plurality of interleaved channels, based on τ_(n) ^(net), S_(net) and Γ_(L), the calculation including calculating a correction factor (g_(matchn)) assuming a matched load for each of the plurality of interleaved channels and calculating a reference correction factor (g_(refl)); applying a respective correction filter of the plurality of correction filters to each of a corresponding plurality of source waveform samples of the waveform generator to produce corrected waveforms; and outputting the corrected waveforms from the plurality of interleaved channels of the arbitrary waveform generator.
 2. A method as in claim 1 wherein each of the plurality of interleaved channels comprises a differential channel.
 3. A method as in claim 1 wherein Γ_(L) is an ideal, calculated value.
 4. A method as in claim 1 wherein Γ_(L) is a measured value.
 5. A method as in claim 1 wherein: an external device is connected to the output port of the arbitrary waveform generator; and the calibration is performed at an output port of the external device.
 6. An arbitrary waveform generator calibrated according to the method of any of claims 1-5. 